[FOM] "Mathematician in the street" on AC

Vaughan Pratt pratt at cs.stanford.edu
Fri Aug 21 18:23:05 EDT 2009

Hi Arnon,

We should agree to agree.  I don't see any of the differences between us 
that you're pointing to as significant.

In particular we agree that these street mathematicians (one pictures 
them performing Hilbert's Nullstellensatz while passers-by drop coins in 
their hat) are enumerating witnesses to countability rather than 
countable sets.  As long as they understand that that's what they're 
enumerating then I don't see the problem.

 > (Indeed, in most cases in which the theorem about the countability
 > of the countable union of countable sets is applied in practice,
 > We have such X and F available, so we do not have to use AC).

Exactly so.  Depriving the street mathematician of her witnesses is like 
depriving a boxer of his fists.  (Hilbert didn't think to apply that 
argument to AC, which he acknowledged as problematic, while applying it 
to intuitionistic logic, where Goedel's translation shows that fists 
only flatten theorems by erasing the distinction between P and ~~P, they 
do not deprive the mathematician of any theorems.)  Why should she care 
that foundationalists make things harder by killing off her witnesses? 
You have to give her a situation she cares about where she has a list of 
countable sets with no witnesses to their countability.  Good luck with 
that, she's probably never run across such a thing.

Unless you can offer her a persuasive example of such, this is what I 
meant in my first post where I said it's probably safer not to bring 
this foundational weirdness up with "working mathematicians" since 
they'll just take it as evidence that foundations creates obstacles 
where none are apparent to the working mathematician.


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